Synthetic division can only be used when dividing by 

x+a or  x-a 

To divide  by x²-1 as in your problem,



you can only use long division:

                2x+5
x²+0x-1)2x³+5x²-4x-1
        2x³+0x²-2x
            5x²-2x-1
            5x²+0x-5
               -2x+4

Answer: 2x + 5 + 

However, you can do the long division synthetically
by writing only the coefficients, like this:

            2  5
1 0 -1)2 5 -4 -1                
       2 0 -2
         5 -2 -1
         5  0 -5
           -2  4

and interpreting the quotient and remainder appropriately
by subtracting the largest exponent of the denominator from
the largest exponent in the numerator and writing this 
exponent of x by the coefficient of the first term in the
quotient. 

Edwin